(No) Exceptional Bundles Over a Curve

Oct 14, 2016

Today we will talk about exceptional objects, or lack thereof, in the derived
category of a smooth and proper curve, say \(C\). Recall, an exceptional
object \(E\) in a k-linear triangulated category \(\mathcal{T}\) is an object such
that
\[
\mathcal{T}(E,E[i])\cong k[0].
\]

During my oral examinations, I was asked about the existence or non-existence of
nontrivial semi-orthogonal decompositions of the derived category of a smooth
and proper curve. Of course \(\mathbb{P}^1\) has Beilinson’s decomposition and
Okawa proves that curves of genus larger than zero have no such decompositions
in Semi-orthogonal decomposability of the Derived
Category of a Curve. They were unsatsified with this answer and asked me to
prove that exceptional objects do not exist if the genus is positive. We will
reprove this fact in this post.

Let \(D(C)\) denote the bounded derived category of coherent sheaves over
\(C\), where \(C\) is a smooth and proper curve of positive genus. This category is
formal since \(C\) is dimension 1. Recall, the derived category is formal
if all objects are formal, i.e. quasi-isomorphic to the cohomology objects.
Instead of looking at complexes of sheaves, we can reduce the question to honest
sheaves. That is, there is no sheaf \(\mathcal{F}\) on \(C\) such that
\(\mathrm{Ext}^\ast_C(\mathcal{F,F})\cong k[0]\).

There are two types of sheaves on a curve \(C\): torsion sheaves and locally
free sheaves. Any sheaf \(\mathcal{F}\) on \(C\) fits into an exact sequence
\[
0\to \mathcal{F}_{tor}\to \mathcal{F}\to \mathcal{E}\to 0
\]
where \(\mathcal{E}\) is locally free and \(\mathcal{F}_{tor}\) is torsion.
Moreover, the relevant extension group vanishes:
\[
\mathrm{Ext}^1_C(\mathcal{E},\mathcal{F}_{tor}) \cong H^1(\mathcal{F}_{tor}\otimes
\mathcal{E}^\vee)\cong H^1(\mathcal{F}_{tor}^{\oplus n}) = 0
\]
where n is the rank of \(\mathcal{E}\). Thus any sheaf \(\mathcal{F}\) on
\(C\) is the sum of a torsion sheaf and a locally free sheaf. It suffices to
show that torsion sheaves are not exceptional and locally free sheaves are not
exceptional.

Let’s start with locally free sheaves. Suppose \(\mathcal{E}\) is a vector
bundle over \(C\), then there is an exact sequence of sheaves
\[
0\to \mathcal{K}\to \mathcal{End}(\mathcal{E})\xrightarrow{\mathrm{Trace}}
\mathcal{O}_C\to 0.
\]

Since \(C\) has positive genus, the mapping
\[
H^1(\mathcal{End}(\mathcal{E}))\to H^1(\mathcal{O}_C)\to 0
\]
is nontrivial and using the isomorphism
\[
H^1(\mathcal{End}(\mathcal{E}))\cong \mathrm{Ext}^1(\mathcal{E},\mathcal{E})
\]
we see that vector bundles cannot be exceptional.

For a torsion sheaf \(\tau\) to be exceptional, it must be supported at
a single closed point, say p. Otherwise,
\(\mathrm{Hom}_C(\tau,\tau)\) is more than one dimensional. As
\(\tau\) is torsion, there is an isomorphism \(\tau\cong
\mathcal{O}_{np}\). By \(\mathcal{O}_{np}\) I mean the cokernel in the
exact sequence
\[
0\to \mathcal{O}(-np)\to \mathcal{O}\to \mathcal{O}_{np}\to 0.
\]
By applying \(\mathbf{R}\mathrm{Hom}_C(-,\mathcal{O}_{np})\), there is a
nontrivial mapping:
\[
0\to \mathrm{Hom}_C(\mathcal{O}(-np),\mathcal{O}_{np})\to
\mathrm{Ext}_C^1(\mathcal{O}_{np},\mathcal{O}_{np}).
\]
It is nontivial because
\(\mathrm{Hom}_C(\mathcal{O}(-np),\mathcal{O}_{np})\neq 0\) and the
isomorphism
\[
\mathrm{Hom}_C(\mathcal{O}_{np},\mathcal{O}_{np})\cong
\mathrm{Hom}_C(\mathcal{O},\mathcal{O}_{np})
\]
induced by pullback.

Thus there are no exceptional torsion sheaves on a curve and this completes the
argument.

This result is a shadow of a more general theme regarding semi-orthogonal
decompositions. In the Fano case, one typically has an exceptional object or an
exceptional collection, then a large piece characterized by the left or right
semi-orthogonal complement. Smooth Fano curves are all isomorphic to
\(\mathbb{P}^1\) and in this case the large piece is not present.

In the smooth and connected Calabi-Yau case, Bridgeland has proven there are no
nontrivial semi-orthogonal decompositions. Roughly, any semi-orthogonal
decomposition must be fully orthogonal by Serre duality. Then this contradicts
connectedness.

In the general type case, they are more rare but interesting examples do exist.
For example, the Beauville surface admits interesting semi-orthogonal
decompositions.